This Christmas several millions of Dogecoins have been stolen from Dogewallet. On Reddit it is explained that the hacker gained access to the filesystem and modified the send/receive page to send to a static address.

Is a similar theft/hack theoretically possible with the (far more valuable) Bitcoins? Or has this happened already in the past?

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The attack was made possible by an error of the coin owners (trusting the online wallet) and another error by the online wallet admins (they didn't make public the exact details, but the attacker certainly exploited a vulnerability that should not have been there).
The same is possible for cash/gold etc: if you trust a safe-deposit box to hold an ounce of gold and a door is accidentally left unlocked, you're pretty much in the same situation as the dogecoin hack.